Large Displacement Behavior Research Articles

Large displacement model for the nonlinear behaviour of the face plate component

This paper deals with the face plate component in steel joints and addresses the full characterization of the nonlinear behaviour of the face plate component. An equivalent beam strip model is proposed that tackles the connection of a beam to the column web of open I-sections in a minor axis joint or the face of tubular columns, covering endplate or fin plate joint typologies. Closed-form analytical solutions are obtained for the elastic large displacement and the elastic-plastic large displacement behaviour of the equivalent beam strip. Criteria for the establishment of the design resistance using the continuous strength method are also proposed. It was concluded that the model is easy to apply, was validated against a large parametric study using finite element beam models, leads to accurate solutions and demonstrates the need to consider membrane effects in design.

A Numerical Study on the Large Displacement in Functionally Graded Beam under Thermal Effect

The large displacement behavior of a Functionally Graded (FG) beam under uniform thermal load is investigated numerically. Six different effects are taken into consideration when examining the large displacement behavior of the beam. These are the effects of temperature, material, geometry, slenderness, force, and boundary condition. The nonlinear numerical analysis is carried out by using the Simulation mode of SolidWorks, which is a finite element-based commercial program. As a result, the displacement of the end of the beam increase with increasing temperature, slenderness ratio, and force. However, it decreases with increase in the ratio of ceramic in Functionally Graded Material (FGM) and the width of the beam. Moreover, as expected, the maximum and minimum displacements are obtained in the beams with Clamped-Free and Clamped-Clamped boundary conditions, respectively.

Size dependent large displacements of microbeams and microframes

The size dependent large displacement behavior of planar microbeams and microframes is studied in this paper using a corotational beam element. To account for the size effect, the modified couple stress theory (MCST) is employed in conjunction with Euler-Bernoulli beam theory in deriving the internal force vector and the tangent stiffness matrix of the beam element. The Newton-Raphson based iterative procedure is used in combination with the arc-length method to solve the nonlinear equilibrium equation and to trace the equilibrium paths. Various microbeams and microframes are analyzed to show the influence of the size effect on the large deflection behavior of the microstructure. The obtained result reveals that the size effect plays an important role on the large deflection response, and the displacements of the structure are over estimated by ignoring the size effect. A parametric study is carried out to highlight the influence of the material length scale parameter on the large displacement behavior of the microbeams and microframes.

Load Bearing Capacity of Light Timber Frame Walls Under Fire

Light Timer Frame Walls are made of solid timber elements and are usually fire protected by other materials. This investigation determines the residual load bearing capacity of Light Timber Frame Walls for fire rating periods of 30, 60, 90 and 120 min. The timber frame is made with 3 studs and 2 tracks, using two different fire protection levels (one and two gypsum layers). The computational model includes the thermal analysis under standard fire, including all types of materials and a sequential mechanical analysis with incremental load applied for each fire rating periods, applied just for the load bearing material. Nonlinear solution methods are used with small tolerance values for solution convergence, to determine the correct temperature and displacement fields. The thermal and the mechanical properties are considered temperature dependent. The mechanical analysis considers large displacement behaviour, and the charring effect of wood is included by the reduction of the stiffness and strength of the timber. The results show that the load bearing capacity decreases with the fire exposure time. A new proposal is presented between the load bearing capacity and the fire rating and finally the char layer is compared with the current and future version of Eurocode 5, part 1.2.

Large displacement behaviour of GRS bridge abutments under passive push

Many bridge abutments suffered failures due to large lateral deformations due to seismic shaking. During earthquakes, bridge abutments restrict the bridge deck's intrusion towards the backfill soil by generating passive earth pressure, which provides resistance to lateral deformation. As a result, the backfill takes the earthquake induced inertial forces transmitted from the bridge superstructure through abutment. Now a days, GRS bridge abutments are widely used due to their many advantages. In this paper, the behaviour of these GRS bridge abutments under large displacements such as 1 m lateral push into the soil due to the impact of bridge deck has been analysed and studied. It is observed that the GRS abutments with longer geogrids are observed to show better passive resistance.

A co-rotational curved beam element for geometrically nonlinear analysis of framed structures

Curved beams are sometimes used in practical framed structures due to good mechanical properties and artistic design. In a framed structure, curved beams may undergo large displacement and experience nonlinear behavior as same as the other straight slender beam-column members. Thus, a geometrically nonlinear curved beam element plays an important role in the analysis of framed structures with curved beams. However, most existing curved beam elements are not accurate enough and still need several or even dozens of elements to accurately describe the behavior of a curved beam with a large subtended angle. To fill this gap, this paper presents a novel geometrically nonlinear curved beam element based on the element-independent co-rotational (EICR) method. The proposed element can simulate a curved beam using only one single element in the analysis and design of most practical framed structures. Moreover, this element is directly derived in a Cartesian coordinate system and can be directly and conveniently used with straight beam-column elements in nonlinear structural analysis. In this manuscript, the derivation of the proposed geometrically nonlinear curved beam element is detailed and several benchmark problems are proposed to verify its accuracy and efficiency.

High-resolution MEMS inertial sensor combining large-displacement buckling behaviour with integrated capacitive readout

Commercially available gravimeters and seismometers can be used for measuring Earth’s acceleration at resolution levels in the order of {mathrm{ng}}/sqrt {mathrm{Hz}} (where g represents earth’s gravity) but they are typically high-cost and bulky. In this work the design of a bulk micromachined MEMS device exploiting non-linear buckling behaviour is described, aiming for {mathrm{ng}}/sqrt {mathrm{Hz}} resolution by maximising mechanical and capacitive sensitivity. High mechanical sensitivity is obtained through low structural stiffness. Near-zero stiffness is achieved through geometric design and large deformation into a region where the mechanism is statically balanced or neutrally stable. Moreover, the device has an integrated capacitive comb transducer and makes use of a high-resolution impedance readout ASIC. The sensitivity from displacement to a change in capacitance was maximised within the design and process boundaries given, by making use of a trench isolation technique and exploiting the large-displacement behaviour of the device. The measurement results demonstrate that the resonance frequency can be tuned from 8.7 Hz–18.7 Hz, depending on the process parameters and the tilt of the device. In this system, which combines an integrated capacitive transducer with a sensitivity of 2.55 aF/nm and an impedance readout chip, the theoretically achievable system resolution equals 17.02 {mathrm{ng}}/sqrt {mathrm{Hz}}. The small size of the device and the use of integrated readout electronics allow for a wide range of practical applications for data collection aimed at the internet of things.

Study on the large-displacement behaviour of a spiral spring with variations of cross-section, orthotropy and prestress

Abstract. This work is dedicated to the study of the large-displacement behaviour of a spiral spring. Parameters that influence the local torsion stiffness of the beam that constitutes the spiral are varied and their effect is studied. Cross-sectional shape, orthotropic material orientation and prestress are the three classes of parameters that are varied. The effect that the local change in torsional stiffness has on the overall behaviour is illustrated in a linearised way by comparing in-plane and out-of-plane stiffnesses, and nonlinearly, by inspecting a graphical representation of the potential energy field of the system. Several embodiments composed of multiple spirals are showed to illustrate how the understanding of the nonlinear behaviour could be exploited in conceptual design of compliant mechanisms.

Formulation of solid-shell finite elements with large displacements considering different transverse shear strains approximations

This work presents a general formulation and implementation in solid-shell elements of the refined zigzag theory and the trigonometric shear deformation theory in an unified way. The model thus conceived is aimed for use in the analysis, design and verification of structures made of composite materials, in which shear strains have a significant prevalence. The refined zigzag theory can deal with composite laminates economically, adding only two nodal degrees of freedom, with very good accuracy. It assumes that the in-plane displacements have a piece-wise linear shape across the thickness depending on the shear stiffness of each composite layer. The trigonometric theory assumes a cosine variation of the transverse shear strain. A modification of this theory is presented in this paper allowing its implementation with C0 approximation functions. Two existing elements are considered, an eight-node tri-linear hexahedron and a six-node triangular prism. Both elements use a modified right Cauchy-Green deformation tensor C¯ where five of its six components are linearly interpolated from values computed at the top and bottom surfaces of the element. The sixth component is computed at the element center and it is enhanced with an additional degree of freedom. This basic kinematic is improved with a hierarchical field of in-plane displacements expressed in convective coordinates. The objective of this approach is to have a simple and efficient finite element formulation to analyze composite laminates under large displacements and rotations but small elastic strains. The assumed natural strain technique is used to prevent transverse shear locking. An analytic through-the-thickness integration and one point integration on the shell plane is used requiring hourglass stabilization for the hexahedral element. Several examples are considered on the one hand to compare with analytical static solutions of plates, and on the other hand to observe natural frequencies, buckling loads and the non-linear large displacement behavior in double curved shells. The results obtained are in a very good agreement with the targets used.

Three-Dimensional Spherical DDA Method for Modeling Friction Problems

AbstractThree-dimensional (3D) discontinuous deformation analysis (DDA) is very suitable for simulating large-displacement behavior of rock masses. One fundamental issue commonly encountered in 3D DDA is 3D friction problems. In this paper, a friction contact model based on 3D spherical DDA (SDDA) and the determination method of friction force direction is proposed for modeling 3D friction problems. Simulation results of three verification examples with available analytical solutions demonstrate the correctness of the friction contact model in 3D SDDA. Meanwhile, the simulation result of one application example, namely, a 3D-sphere natural-accumulation simulation, indicates that the friction problems of a 3D-sphere system can be modeled well by the proposed friction contact model. This work is helpful for solving 3D friction problems in discrete blocky systems.

Large displacement behavior of double parallelogram flexure mechanisms with underconstraint eliminators

This paper presents a new analytical method for predicting the large displacement behavior of flexural double parallelogram (DP) bearings with underconstraint eliminator (UE) linkages. This closed-form perturbative Euler analysis method is able to – for the first time – directly incorporate the elastomechanics of a discrete UE linkage, which is a hybrid flexure element that is linked to ground as well as both stages on the bearing. The models are used to understand a nested linkage UE design, however the method is extensible to other UE linkages. Design rules and figures-of-merit are extracted from the analysis models, which provide powerful tools for accelerating the design process. The models, rules and figures-of-merit enable the rapid design of a UE for a desired large displacement behavior, as well as providing a means for determining the limits of UE and DP structures. This will aid in the adoption of UE linkages into DP bearings for precision mechanisms. Models are generated for a nested linkage UE design, and the performance of this DP with UE structure is compared to a DP-only bearing. The perturbative Euler analysis is shown to match existing theories for DP-only bearings with distributed compliance within ≈2%, and Finite Element Analysis for the DP with UE bearings within an average 10%.

Validation of flexible multibody dynamics beam formulations using benchmark problems

As the need to model flexibility arose in multibody dynamics, the floating frame of reference formulation was developed, but this approach can yield inaccurate results when elastic displacements becomes large. While the use of three-dimensional finite element formulations overcomes this problem, the associated computational cost is overwhelming. Consequently, beam models, which are one-dimensional approximations of three-dimensional elasticity, have become the workhorse of many flexible multibody dynamics codes. Numerous beam formulations have been proposed, such as the geometrically exact beam formulation or the absolute nodal coordinate formulation, to name just two. New solution strategies have been investigated as well, including the intrinsic beam formulation or the DAE approach. This paper provides a systematic comparison of these various approaches, which will be assessed by comparing their predictions for four benchmark problems. The first problem is the Princeton beam experiment, a study of the static large displacement and rotation behavior of a simple cantilevered beam under a gravity tip load. The second problem, the four-bar mechanism, focuses on a flexible mechanism involving beams and revolute joints. The third problem investigates the behavior of a beam bent in its plane of greatest flexural rigidity, resulting in lateral buckling when a critical value of the transverse load is reached. The last problem investigates the dynamic stability of a rotating shaft. The predictions of eight independent codes are compared for these four benchmark problems and are found to be in close agreement with each other and with experimental measurements, when available.

A Corotational Formulation for Large Displacement Analysis of Functionally Graded Sandwich Beam and Frame Structures

A corotational finite element formulation for large displacement analysis of planar functionally graded sandwich (FGSW) beam and frame structures is presented. The beams and frames are assumed to be formed from a metallic soft core and two symmetric functionally graded skin layers. The Euler-Bernoulli beam theory and von Kármán nonlinear strain-displacement relationship are adopted for the local strain. Exact solution of nonlinear equilibrium equations for a beam segment is employed to interpolate the displacement field for avoiding the membrane locking. An incremental-iterative procedure is used in combination with the arc-length control method to compute the equilibrium paths. Numerical examples show that the proposed formulation is capable of evaluating accurately the large displacement response with just several elements. A parametric study is carried out to highlight the effect of the material distribution, the core thickness to height ratio on the large displacement behaviour of the FGSW beam, and frame structures.

Curling of a Heated Annulus

We study the large-displacement behavior of a thin annulus subjected to thermal gradients. Following axisymmetrical deflections, the annulus buckles into a periodic mode shape, which eventually forms a number of a radial curls under sustained heating. We predict the number of curls using a simple plate-buckling analogy and an approximate energy method.

The isolation characteristics of an archetypal dynamical model with stable-quasi-zero-stiffness

In this paper, a single-degree-of-freedom geometrically nonlinear oscillator with stable-quasi-zero-stiffness (SQZS) is presented, which can be extensively applied in vibration isolation due to its high static load bearing capacity and low dynamic stiffness. This model comprises a lumped mass denoeing the isolated object and a pair of horizontal springs providing negative stiffness paralleled with a vertical linear spring to bear the load. The equation of motion of the system is formulated with an originally irrational nonlinearity based upon SD oscillator instead of the conventionally approximate Duffing system of polynomial type, which will produce results with a high precision unquestionably, especially for the prediction of a large displacement behaviour. The frequency response characteristics particularly for transmissibility of the model, subjected to harmonic forcing and vibrating base, are obtained by using an extended averaging approach to achieve the parameter optimization for maximum frequency band of isolation. Furthermore, numerical simulations are carried out to detect the complex dynamical phenomena of periodic, chaotic motions and coexistence of multiple solutions, and so on, in addition to verifying the analytical results. Finally, an interesting strategy is proposed to extend the frequency band of isolation by controlling the initial value within the attraction basin of a small amplitude attractor, which can be utilized for an effective isolation. The results presented herein provide an insight of dynamics into the SQZS mechanism for its application in vibration engineering.

Large displacement behavior of a rocking flexible structure under harmonic excitation

In this paper the nonlinear dynamic behavior of a rocking flexible structure subjected to harmonic excitation is investigated. The dynamic response of the structure is characterized by a sequence of full contact and rocking phases which may be halted by the overturning of the system. In the study, reference is made to a simple lumped mass model which, in the full contact phase, behaves as a linear single degree-of-freedom system. In the rocking phase the system exhibits two degrees of freedom and its response is evaluated with reference to large rigid rotations and small elastic deformations. New explicit and numerical results are presented as a function of structural and loading parameters. The complex dynamic behavior of the rocking flexible structure, subjected to harmonic excitation, is compared with the corresponding nonlinear dynamics of the rigid system with the aim of highlighting the limits of applicability of the rigid body assumption. Extensive numerical applications have been performed in order to evaluate the main features of the nonlinear response of the flexible model subjected to harmonic acceleration at the base.

Large displacement behaviour of tapered cantilever Euler–Bernoulli beams made of functionally graded material

The large displacement behaviour of tapered cantilever Euler–Bernoulli beams made of functionally graded material subjected to end forces is studied by the finite element method. The effective Young’s modulus of the beams is assumed to be graded in the thickness direction by a power-law distribution. Based on the co-rotational approach, a finite element formulation is derived and employed in the study. An incremental/iterative procedure in combination with the arc-length control method is used in computing the large displacement response of the beams. The numerical results show that the derived formulation is capable to give accurate results by using just several elements. A parametric study is given to highlight the influence of the material inhomogeneity, taper ratio and taper type on the large displacement behaviour of the beams. The large displacement behaviour of beams composed of different constituent materials is also investigated and highlighted.

Analysis of a Compliant Mechanism for Positioning in the cm‐Range

AbstractThe kinematics and kinetics of a compliant mechanism are analyzed quasi‐statically. The mechanism consists of a parallelkinematic (Biglide type) with the particular feature, that its revolute joints are implemented by flexure hinges. In the considered example the flexure hinges are uniform beams similar to leaf springs. The preliminary considerations with a rigid body system and by linear theory of elasticity result in an initial geometry. In order to predict the large displacement behavior more precisely the theory of Elastica is used. An optimization method is applied to solve the mechanism's state of deformation. It was found that some aspects can be approximated by a pseudo rigid body system, while others, namely the rotation, cannot be rendered. The decreased stiffness in the highly deformed operating state appears as key problem. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Morphing of curved corrugated shells

Thin sheet materials of low bending stiffness but high membrane stiffness are often corrugated in order to achieve improvements of several orders of magnitude in bending stiffness with only minimal increases in weight and cost. If these corrugated sheets are initially curved along the corrugations, much of this stiffness gain is lost. In return, the sheets are then capable of significant elastic changes in shape overall, including large changes in overall Gaussian curvature. These shape changes are described here by non-linear and coupled kinematical relationships, which are verified against experiment and finite-element simulations. It is found that gross simplifications can be made about the large displacement behaviour of such shells without a loss of accuracy.

Implementation of advanced analysis method for steel-framed structures under fire conditions

This paper is concerned with the development of an advanced analysis numerical tool capable of estimating the inelastic large-displacement behaviour of plane steel-framed structures under fire conditions. A non-linear transient heat transfer analysis is performed on the basis of the finite element method (FEM), following the main guidelines proposed by the European Code for steel structures under fire conditions. The computational analysis program is used to assess the structural load-bearing functions and to estimate the structural behaviour and the corresponding time-resistance period. The original refined plastic hinge method is extended for fire design analysis considering both tangent modulus model and inelastic stiffness degradation concepts in the developed computational program. A tangent modulus model is developed for the European column buckling-curve for fire condition. A gradual inelastic surface is proposed as a function of the temperature and the loading combination. The results obtained are compared with those from an FEM computational program and those from the Eurocode simplified design recommendations. The benefits of using steel ductility in the current design of fire-unprotected structures are outlined.